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This is lecture twelve.
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OK.
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We've reached twelve lectures.
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And this one is more than
the others about applications
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of linear algebra.
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And I'll confess.
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When I'm giving you examples
of the null space and the row
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space, I create a little matrix.
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You probably see that I
just invent that matrix
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as I'm going.
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And I feel a little
guilty about it,
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because the truth is that real
linear algebra uses matrices
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that come from somewhere.
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They're not just, like, randomly
invented by the instructor.
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They come from applications.
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They have a definite structure.
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And anybody who works with
them gets, uses that structure.
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I'll just report,
like, this weekend
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I was at an event with
chemistry professors.
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OK, those guys are row
reducing matrices, and what
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matrices are they working with?
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Well, their little matrices tell
them how much of each element
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goes into the --
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or each molecule, how
many molecules of each
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go into a reaction
and what comes out.
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And by row reduction they
get a clearer picture
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of a complicated reaction.
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And this weekend I'm going to --
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to a sort of birthday
party at Mathworks.
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So Mathworks is out
Route 9 in Natick.
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That's where Matlab is created.
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It's a very, very
successful, software,
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tremendously successful.
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And the conference will be about
how linear algebra is used.
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And so I feel
better today to talk
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about what I think is
the most important model
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in applied math.
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And the discrete
version is a graph.
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So can I draw a graph?
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Write down the matrix
that's associated with it,
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and that's a great
source of matrices.
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You'll see.
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So a graph is
just, so a graph --
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to repeat -- has
nodes and edges.
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OK.
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And I'm going to write down
the graph, a graph, so I'm just
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creating a small graph here.
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As I mentioned last time,
we would be very interested
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in the graph of all, websites.
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Or the graph of all telephones.
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I mean -- or the graph of
all people in the world.
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Here let me take just,
maybe nodes one two three --
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well, I better put in an --
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I'll put in that edge and maybe
an edge to, to a node four,
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and another edge to node four.
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How's that?
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So there's a graph
with four nodes.
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So n will be four in my --
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n equal four nodes.
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And the matrix will have
m equal the number --
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there'll be a row
for every edge,
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so I've got one two
three four five edges.
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So that will be
the number of rows.
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And I have to to write down
the matrix that I want to,
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I want to study, I need to
give a direction to every edge,
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so I know a plus and
a minus direction.
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So I'll just do
that with an arrow.
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Say from one to two, one
to three, two to three,
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one to four, three to four.
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That just tells me, if I have
current flowing on these edges
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then I know whether it's
-- to count it as positive
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or negative according as whether
it's with the arrow or against
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the arrow.
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But I just drew those
arrows arbitrarily.
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OK.
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Because I -- my example is going
to come -- the example I'll --
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the words that I will use
will be words like potential,
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potential difference, currents.
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In other words, I'm
thinking of an electrical
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network.
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But that's just one possibility.
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My applied math class
builds on this example.
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It could be a hydraulic
network, so we could be doing,
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flow of water, flow of oil.
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Other examples, this
could be a structure.
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Like the -- a design
for a bridge or a design
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for a Buckminster Fuller dome.
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Or many other
possibilities, so many.
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So l- but let's take
potentials and currents
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as, as a basic
example, and let me
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create the matrix that tells
you exactly what the graph tells
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you.
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That's --, that's it.
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So now I'll call it the
incidence matrix, incidence
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matrix.
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So let me write it
down, and you'll see,
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OK.
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what its properties are.
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So every row
corresponds to an edge.
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I have five rows
from five edges,
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and let me write down again
what this graph looks like.
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OK, the first edge, edge one,
goes from node one to two.
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So I'm going to put in a minus
one and a plus one in th- this
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corresponds to node
one two three and four,
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That's a basis for the null
space. the four columns.
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The five rows correspond -- the
first row corresponds to edge
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one.
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Edge one leaves node one and
goes into node two, and that --
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and it doesn't touch
three and four.
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Edge two, edge two goes -- oh,
I haven't numbered these edges.
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I just figured that was probably
edge one, but I didn't say so.
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Let me take that to be edge one.
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Let me take this to be edge two.
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Let me take this
to be edge three.
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This is edge four.
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Ho, I'm discovering
-- no, wait a minute.
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Did I number that twice?
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Here's edge four.
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And here's edge five.
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All right.
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OK?
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So, so edge one, as I said,
goes from node one to two.
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Edge two goes from two to
three, node two to three, so
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minus one and one in the
second and third columns.
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Edge three goes
from one to three.
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I'm, I'm tempted to stop for a
moment with those three edges.
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The null space is actually one
dimensional, and it's the line
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Edges one two three,
those form what would we,
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A basis for the
null space will be
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just that.1 what do you call
the, the little, the little,
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the subgraph formed by
edges one, two, and three?
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That's a loop.
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And the number of loops and
the position of the loops
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will be crucial.
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OK.
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Actually, here's a
interesting point about loops.
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If I look at those
rows, corresponding
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to edges one two three,
and these guys made a loop.
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You want to tell me --
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if I just looked at
that much of the matrix
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it would be natural
for me to ask,
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are those rows independent?
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Are the rows independent?
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And can you tell from looking
at that if they are or are not
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independent?
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Do you see a, a relation
between those three rows?
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Yes.
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If I add that row to
that row, I get this row.
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So, so that's like a hint
here that loops correspond
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to dependent, linearly
dependent column --
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linearly dependent give me
a basis for the null space.
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rows.
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OK, let me complete
the incidence matrix.
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Number four, edge four is going
from node one to node four.
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And the fifth edge is going
from node three to node four.
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OK.
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There's my matrix.
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It came from the five
edges and the four nodes.
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And if I had a big graph,
I'd have a big matrix.
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And what questions do
I ask about matrices?
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Can I ask --
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here's the review now.
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There's a matrix that comes
from somewhere. of all vectors
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through that one.
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If, if it was a big graph,
it would be a large matrix,
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but a lot of zeros, right?
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Because every row only
has two non-zeros.
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So the number of -- it's
a very sparse matrix.
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The number of non-zeros
is exactly two times
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five, it's two m.
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Every row only
has two non-zeros.
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And that's with a
lot of structure.
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And -- that was the point
I wanted to begin with,
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that graphs, that real
graphs from real --
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real matrices from genuine
problems have structure.
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OK.
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We can ask, and because of
the structure, we can answer,
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If it -- yeah, let me
ask you just always,
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the, the main questions
about matrices.
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So first question, what
about the null space?
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So what I asking if I ask you
for the null space of that
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matrix?
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I'm asking you if I'm looking
at the columns of the matrix,
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four columns, and I'm asking
you, So there's a basis for it,
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and here is the whole null
are those columns independent?
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If the columns are independent,
then what's in the null space?
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Only the zero vector, right?
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The null space contains --
tells us what combinations
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of the columns -- it tells us
how to combine columns to get
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zero.
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Can -- and is there anything in
the null space of this matrix
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other than just the zero vector?
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In other words, are
those four columns
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independent or dependent?
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What else is in the null space?
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OK.
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That's our question.
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Let me, I don't know
if you see the answer.
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Whether there's -- so let's see.
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I guess we could do it properly.
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space.
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We could solve Ax=0.
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So let me solve Ax=0
to find the null space.
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OK.
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What's Ax?
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Can I put x in here
in, in little letters?
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x1, x2, x3, x4, that's
-- it's got four columns.
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Ax now is that matrix times x.
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And what do I get for Ax?
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If the camera can keep that
matrix multiplication there,
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I'll put the answer here.
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Ax equal -- what's the
first component of Ax?
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Can you take that first row,
minus one one zero zero,
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and multiply by the x, and
of course you get x2-x1.
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space.
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The second row, I get x3-x2.
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From the third row, I get x3-x1.
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Any multiple of one one one
one, it's the whole line in four
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From the fourth
row, I get x4-x1.
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And from the fifth
row, I get x4-x3.
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And I want to know
when is the thing zero.
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This is my equation, Ax=0.
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Notice what that
matrix A is doing,
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what we've created a
matrix that computes
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the differences across
every edge, the differences
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in potential. differences are
zero, and that x is in the null
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Let me even begin to
give this interpretation.
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I'm going to think of this
vector x, which is x1 x2 x3 x4,
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as the potentials at the nodes.
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So I'm introducing a word,
potentials at the nodes.
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And now if I multiply
by A, I get these --
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I get these five components,
x2-x1, et cetera.
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And what are they?
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They're potential differences.
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That's what A computes.
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If I have potentials at the
nodes and I multiply by A,
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it gives me the potential
differences, the differences
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in potential, across the edges.
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OK.
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When are those
differences all zero?
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So I'm looking for
the null space.
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Of course, if all the (x)s
are zero then I get zero.
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That, that just tells me,
of course, the zero vector
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is in the null space.
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But w- there's more
in the null space.
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Those columns are -- of
A are dependent, right --
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because I can find
solutions to that equation.
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dimensional space.
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Tell me -- the null space.
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Tell me one vector in the
null space, so tell me an x,
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it's got four components,
and it makes that thing zero.
252
00:15:10,490 --> 00:15:13,890
So what's a good x to do that?
253
00:15:13,890 --> 00:15:22,000
One one one one,
constant potential.
254
00:15:22,000 --> 00:15:52,130
If the potentials are constant,
then all the potential
255
00:15:52,130 --> 00:16:03,430
Do you see that
that's the null space?
256
00:16:03,430 --> 00:16:12,260
So the, so the dimension of
the null space of A is one.
257
00:16:12,260 --> 00:16:14,690
And there's a basis
for it, and there's
258
00:16:14,690 --> 00:16:16,190
everything that's in it.
259
00:16:16,190 --> 00:16:17,000
Good.
260
00:16:17,000 --> 00:16:23,250
And what does that
mean physically?
261
00:16:23,250 --> 00:16:25,240
I mean, what does that
mean in the application?
262
00:16:25,240 --> 00:16:26,850
That guy in the null space.
263
00:16:26,850 --> 00:16:33,170
It means that the
potentials can only
264
00:16:33,170 --> 00:16:36,560
be determined up to a constant.
265
00:16:36,560 --> 00:16:39,560
Potential differences are
what make current flow.
266
00:16:39,560 --> 00:16:40,970
That's what makes things happen.
267
00:16:45,360 --> 00:16:47,140
It's these potential
differences that
268
00:16:47,140 --> 00:16:50,320
will make something move
in the, in our network,
269
00:16:50,320 --> 00:16:53,950
between x2- between
node two and node one.
270
00:16:53,950 --> 00:16:57,650
Nothing will move if all
potentials are the same.
271
00:16:57,650 --> 00:17:01,710
If all potentials are c, c, c,
and c, then nothing will move.
272
00:17:01,710 --> 00:17:06,190
So we're, we have
this one parameter,
273
00:17:06,190 --> 00:17:10,060
this arbitrary constant
that raises or drops
274
00:17:10,060 --> 00:17:11,359
all the potentials.
275
00:17:11,359 --> 00:17:14,339
It's like ranking
football teams, whatever.
276
00:17:14,339 --> 00:17:16,510
We have a, there's a,
there's a constant --
277
00:17:16,510 --> 00:17:20,540
or looking at
temperatures, you know,
278
00:17:20,540 --> 00:17:23,770
there's a flow of heat from
higher temperature to lower
279
00:17:23,770 --> 00:17:26,349
temperature.
280
00:17:26,349 --> 00:17:28,840
If temperatures are
equal there's no flow,
281
00:17:28,840 --> 00:17:30,700
and therefore we can measure --
282
00:17:30,700 --> 00:17:36,950
we can measure temperatures
by, Celsius or we
283
00:17:36,950 --> 00:17:40,050
can start at absolute zero.
284
00:17:40,050 --> 00:17:44,000
And that arbitrary -- it's the
same arbitrary constant that,
285
00:17:44,000 --> 00:17:46,660
that was there in calculus.
286
00:17:46,660 --> 00:17:49,030
In calculus, right,
when you took
287
00:17:49,030 --> 00:17:53,050
the integral, the indefinite
integral, there was a plus c,
288
00:17:53,050 --> 00:17:58,030
and you had to set a starting
point to know what that c was.
289
00:17:58,030 --> 00:18:02,370
So here what often happens is
we fix one of the potentials,
290
00:18:02,370 --> 00:18:06,370
like the last one.
291
00:18:06,370 --> 00:18:12,050
So a typical thing would
be to ground that node.
292
00:18:12,050 --> 00:18:16,040
To set its potential at zero.
293
00:18:16,040 --> 00:18:19,070
And if we do that, if
we fix that potential
294
00:18:19,070 --> 00:18:25,570
so it's not unknown anymore,
then that column disappears
295
00:18:25,570 --> 00:18:29,440
and we have three columns,
and those three columns
296
00:18:29,440 --> 00:18:30,420
are independent.
297
00:18:30,420 --> 00:18:33,630
So I'll leave the
column in there,
298
00:18:33,630 --> 00:18:35,950
but we'll remember
that grounding a node
299
00:18:35,950 --> 00:18:38,520
is the way to get it out.
300
00:18:38,520 --> 00:18:42,680
And grounding a node is the
way to -- setting a node --
301
00:18:42,680 --> 00:18:47,890
setting a potential to zero
tells us the, the base for all
302
00:18:47,890 --> 00:18:48,510
potentials.
303
00:18:48,510 --> 00:18:51,370
Then we can compute the others.
304
00:18:51,370 --> 00:18:55,070
But what's the -- now
I've talked enough to ask
305
00:18:55,070 --> 00:18:58,420
OK. what the rank
of the matrix is?
306
00:18:58,420 --> 00:19:01,240
What's the rank then?
307
00:19:01,240 --> 00:19:03,000
The rank of the matrix.
308
00:19:03,000 --> 00:19:06,030
So we have a five
by four matrix.
309
00:19:06,030 --> 00:19:11,680
We've located its null
space, one dimensional.
310
00:19:11,680 --> 00:19:13,870
How many independent
columns do we have?
311
00:19:13,870 --> 00:19:15,050
What's the rank?
312
00:19:15,050 --> 00:19:17,880
It's three.
313
00:19:17,880 --> 00:19:21,180
And the first three columns,
or actually any three columns,
314
00:19:21,180 --> 00:19:22,920
will be independent.
315
00:19:22,920 --> 00:19:28,800
Any three potentials are
independent, good variables.
316
00:19:28,800 --> 00:19:35,490
The fourth potential
is not, we need to set,
317
00:19:35,490 --> 00:19:38,160
and typically we
ground that node.
318
00:19:38,160 --> 00:19:38,660
OK.
319
00:19:38,660 --> 00:19:40,270
Rank is three.
320
00:19:40,270 --> 00:19:43,820
Rank equals three.
321
00:19:43,820 --> 00:19:45,200
OK.
322
00:19:45,200 --> 00:19:48,420
Let's see, do I want to ask
you about the column space?
323
00:19:48,420 --> 00:19:52,360
The column space is all
combinations of those columns.
324
00:19:52,360 --> 00:19:55,560
I could say more
about it and I will.
325
00:19:55,560 --> 00:20:01,870
Let me go to the null
space of A transpose,
326
00:20:01,870 --> 00:20:07,680
because the equation A
transpose y equals zero
327
00:20:07,680 --> 00:20:10,090
is probably the most
fundamental equation
328
00:20:10,090 --> 00:20:11,410
of applied mathematics.
329
00:20:11,410 --> 00:20:14,860
All right, let's
talk about that.
330
00:20:14,860 --> 00:20:17,230
That deserves our attention.
331
00:20:17,230 --> 00:20:21,590
A transpose y equals zero.
332
00:20:21,590 --> 00:20:29,510
Let's -- let me put it on here.
333
00:20:29,510 --> 00:20:32,991
So A transpose y equals zero.
334
00:20:32,991 --> 00:20:33,490
OK.
335
00:20:33,490 --> 00:20:38,360
So now I'm finding the
null space of A transpose.
336
00:20:38,360 --> 00:20:41,200
Oh, and if I ask
you its dimension,
337
00:20:41,200 --> 00:20:43,760
you could tell me what it is.
338
00:20:43,760 --> 00:20:49,140
What's the dimension of the
null space of A transpose?
339
00:20:49,140 --> 00:20:51,640
We now know enough to
answer that question.
340
00:20:51,640 --> 00:20:55,030
What's the general formula for
the dimension of the null space
341
00:20:55,030 --> 00:20:55,770
of A transpose?
342
00:20:58,760 --> 00:21:02,560
A transpose, let me even
write out A transpose.
343
00:21:02,560 --> 00:21:10,070
This A transpose will
be n by m, right?
344
00:21:10,070 --> 00:21:12,770
In this case, it'll
be four by five.
345
00:21:12,770 --> 00:21:13,960
n by m.
346
00:21:13,960 --> 00:21:16,320
Those columns will become rows.
347
00:21:16,320 --> 00:21:25,480
Minus one zero minus one minus
one zero is now the first row.
348
00:21:25,480 --> 00:21:31,710
The second row of the matrix,
one minus one and three zeros.
349
00:21:31,710 --> 00:21:36,210
The third column now becomes
the third row, zero one one
350
00:21:36,210 --> 00:21:38,480
zero minus one.
351
00:21:38,480 --> 00:21:41,905
And the fourth column
becomes the fourth row.
352
00:21:45,730 --> 00:21:46,560
OK, good.
353
00:21:46,560 --> 00:21:48,400
There's A transpose.
354
00:21:48,400 --> 00:21:55,965
That multiplies y,
y1 y2 y3 y4 and y5.
355
00:22:00,740 --> 00:22:01,410
OK.
356
00:22:01,410 --> 00:22:03,830
Now you've had time to
think about this question.
357
00:22:03,830 --> 00:22:09,370
What's the dimension of the
null space, if I set all those
358
00:22:09,370 --> 00:22:10,705
-- wow.
359
00:22:13,410 --> 00:22:15,960
Usually -- sometime
during this semester,
360
00:22:15,960 --> 00:22:19,570
I'll drop one of these
erasers behind there.
361
00:22:19,570 --> 00:22:20,880
That's a great moment.
362
00:22:20,880 --> 00:22:22,570
There's no recovery.
363
00:22:22,570 --> 00:22:29,390
There's -- centuries of
erasers are back there.
364
00:22:29,390 --> 00:22:29,890
OK.
365
00:22:35,100 --> 00:22:38,020
OK, what's the dimension
of the null space?
366
00:22:40,640 --> 00:22:42,510
Give me the general
formula first
367
00:22:42,510 --> 00:22:46,050
in terms of r and m and n.
368
00:22:46,050 --> 00:22:49,580
This is like crucial, you --
369
00:22:49,580 --> 00:22:54,200
we struggled to, to decide
what dimension meant,
370
00:22:54,200 --> 00:22:59,980
and then we figured out
what it equaled for an m
371
00:22:59,980 --> 00:23:05,850
by n matrix of rank r, and
the answer was m-r, right?
372
00:23:05,850 --> 00:23:14,200
There are m=5 components,
m=5 columns of A transpose.
373
00:23:14,200 --> 00:23:18,400
And r of those columns
are pivot columns,
374
00:23:18,400 --> 00:23:19,960
because it'll have r pivots.
375
00:23:19,960 --> 00:23:21,410
It has rank r.
376
00:23:21,410 --> 00:23:28,090
And m-r are the free
ones now for A transpose,
377
00:23:28,090 --> 00:23:32,060
so that's five minus
three, so that's two.
378
00:23:35,040 --> 00:23:39,400
And I would like to
find this null space.
379
00:23:39,400 --> 00:23:41,800
I know its dimension.
380
00:23:41,800 --> 00:23:45,440
Now I want to find
out a basis for it.
381
00:23:45,440 --> 00:23:48,780
And I want to understand
what this equation is.
382
00:23:48,780 --> 00:23:53,500
So let me say what A transpose
y actually represents, why I'm
383
00:23:53,500 --> 00:23:57,120
interested in that equation.
384
00:23:57,120 --> 00:24:04,090
I'll put it down with those
old erasers and continue this.
385
00:24:04,090 --> 00:24:07,430
Here's the great picture
of applied mathematics.
386
00:24:07,430 --> 00:24:09,560
So let me complete that.
387
00:24:09,560 --> 00:24:14,040
There's a matrix
that I'll call C
388
00:24:14,040 --> 00:24:17,830
that connects potential
differences to currents.
389
00:24:17,830 --> 00:24:21,840
So I'll call these -- these
are currents on the edges,
390
00:24:21,840 --> 00:24:27,990
y1 y2 y3 y4 and y5.
391
00:24:27,990 --> 00:24:30,340
Those are currents on the edges.
392
00:24:34,160 --> 00:24:39,090
And this relation between
current and potential
393
00:24:39,090 --> 00:24:41,820
difference is Ohm's Law.
394
00:24:41,820 --> 00:24:43,340
This here is Ohm's Law.
395
00:24:47,060 --> 00:24:50,300
Ohm's Law says that
the current on an edge
396
00:24:50,300 --> 00:24:56,080
is some number times
the potential drop.
397
00:24:56,080 --> 00:24:59,820
That's -- and that number is
the conductance of the edge,
398
00:24:59,820 --> 00:25:01,290
one over the resistance.
399
00:25:01,290 --> 00:25:08,850
This is the old current
is, is, the relation
400
00:25:08,850 --> 00:25:13,960
of current, resistance,
and change in potential.
401
00:25:13,960 --> 00:25:17,760
So it's a change in potential
that makes some current happen,
402
00:25:17,760 --> 00:25:22,170
and it's Ohm's Law that says
how much current happens.
403
00:25:22,170 --> 00:25:22,820
OK.
404
00:25:22,820 --> 00:25:25,940
And then the final
step of this framework
405
00:25:25,940 --> 00:25:31,070
is the equation A
transpose y equals zero.
406
00:25:33,950 --> 00:25:38,990
And that's -- what
is that saying?
407
00:25:38,990 --> 00:25:40,470
It has a famous name.
408
00:25:40,470 --> 00:25:50,590
It's Kirchoff's Current Law,
KCL, Kirchoff's Current Law,
409
00:25:50,590 --> 00:25:53,030
A transpose y equals zero.
410
00:25:53,030 --> 00:25:55,950
So that when I'm solving, and
when I go back up with this
411
00:25:55,950 --> 00:26:03,770
blackboard and solve A
transpose y equals zero,
412
00:26:03,770 --> 00:26:05,950
it's this pattern of --
413
00:26:05,950 --> 00:26:08,110
that I want you to see.
414
00:26:08,110 --> 00:26:11,360
That we had rectangular
matrices, but --
415
00:26:11,360 --> 00:26:17,070
and real applications, but in
those real applications comes A
416
00:26:17,070 --> 00:26:18,500
and A transpose.
417
00:26:18,500 --> 00:26:22,030
So our four subspaces are
exactly the right things
418
00:26:22,030 --> 00:26:23,461
to know about.
419
00:26:23,461 --> 00:26:23,960
All right.
420
00:26:23,960 --> 00:26:28,030
Let's know about that
null space of A transpose.
421
00:26:28,030 --> 00:26:31,990
Wait a minute, where'd it go?
422
00:26:31,990 --> 00:26:33,190
There it is.
423
00:26:33,190 --> 00:26:34,680
OK.
424
00:26:34,680 --> 00:26:35,720
OK.
425
00:26:35,720 --> 00:26:38,210
Null space of A transpose.
426
00:26:38,210 --> 00:26:40,070
We know what its
dimension should be.
427
00:26:43,340 --> 00:26:47,670
Let's find out -- tell
me a vector in it.
428
00:26:47,670 --> 00:26:50,150
Tell me -- now, so
what I asking you?
429
00:26:50,150 --> 00:26:53,850
I'm asking you for
five currents that
430
00:26:53,850 --> 00:26:57,320
satisfy Kirchoff's Current Law.
431
00:26:57,320 --> 00:26:59,650
So we better understand
what that law says.
432
00:26:59,650 --> 00:27:01,780
That, that law, A
transpose y equals
433
00:27:01,780 --> 00:27:09,430
zero, what does that say, say
in the first row of A transpose?
434
00:27:09,430 --> 00:27:13,100
That says -- the so the first
row of A transpose says minus
435
00:27:13,100 --> 00:27:19,280
y1 minus y3 minus y4 is zero.
436
00:27:22,740 --> 00:27:25,020
Where did that
equation come from?
437
00:27:25,020 --> 00:27:27,470
Let me -- I'll redraw the graph.
438
00:27:27,470 --> 00:27:31,600
Can I redraw the graph here,
so that we -- maybe here,
439
00:27:31,600 --> 00:27:34,900
so that we see again --
440
00:27:34,900 --> 00:27:39,470
there was node one,
node two, node three,
441
00:27:39,470 --> 00:27:41,750
node four was off here.
442
00:27:41,750 --> 00:27:45,310
That was, that was our graph.
443
00:27:45,310 --> 00:27:47,220
We had currents on those.
444
00:27:47,220 --> 00:27:50,650
We had a current y1 going there.
445
00:27:50,650 --> 00:27:53,120
We had a current y --
what were the other,
446
00:27:53,120 --> 00:27:58,900
what are those edge numbers?
y4 here and y3 here.
447
00:28:01,480 --> 00:28:04,990
And then a y2 and a y5.
448
00:28:04,990 --> 00:28:07,860
I'm, I'm just copying what
was on the other board
449
00:28:07,860 --> 00:28:10,470
so it's ea-
convenient to see it.
450
00:28:10,470 --> 00:28:15,260
What is this equation telling
me, this first equation
451
00:28:15,260 --> 00:28:19,230
of Kirchoff's Current Law?
452
00:28:19,230 --> 00:28:21,900
What does that mean
for that graph?
453
00:28:21,900 --> 00:28:29,320
Well, I see y1, y3, and y4 as
the currents leaving node one.
454
00:28:29,320 --> 00:28:32,930
So sure enough, the first
equation refers to node one,
455
00:28:32,930 --> 00:28:34,720
and what does it say?
456
00:28:34,720 --> 00:28:39,430
It says that the
net flow is zero.
457
00:28:39,430 --> 00:28:43,230
That, that equation A transpose
y, Kirchoff's Current Law,
458
00:28:43,230 --> 00:28:47,860
is a balance equation,
a conservation law.
459
00:28:47,860 --> 00:28:51,770
Physicists, be overjoyed,
right, by this stuff.
460
00:28:51,770 --> 00:28:56,520
It, it says that in equals out.
461
00:28:56,520 --> 00:29:01,770
And in this case, the three
arrows are all going out,
462
00:29:01,770 --> 00:29:05,070
so it says y1, y3,
and y4 add to zero.
463
00:29:05,070 --> 00:29:07,280
Let's take the next one.
464
00:29:07,280 --> 00:29:16,150
The second row is y1-y2, and
that's all that's in that row.
465
00:29:16,150 --> 00:29:19,790
And that must have something
to do with node two.
466
00:29:19,790 --> 00:29:25,780
And sure enough, it says y1=y2,
current in equals current out.
467
00:29:25,780 --> 00:29:33,300
The third one, y2 plus
y3 minus y5 equals
468
00:29:33,300 --> 00:29:34,200
zero.
469
00:29:34,200 --> 00:29:38,340
That certainly will be
what's up at the third node.
470
00:29:38,340 --> 00:29:43,350
y2 coming in, y3 coming in,
y5 going out has to balance.
471
00:29:43,350 --> 00:29:48,770
And finally, y4
plus y5 equals zero
472
00:29:48,770 --> 00:29:57,760
says that at this node, y4
plus y5, the total flow,
473
00:29:57,760 --> 00:30:01,530
We don't -- you know, charge
doesn't accumulate at is zero.
474
00:30:01,530 --> 00:30:03,000
the nodes.
475
00:30:03,000 --> 00:30:06,180
It travels around.
476
00:30:06,180 --> 00:30:07,030
OK.
477
00:30:07,030 --> 00:30:09,940
Now give me --
478
00:30:09,940 --> 00:30:12,580
I come back now to the
linear algebra question.
479
00:30:12,580 --> 00:30:17,310
What's a vector y that
solves these equations?
480
00:30:17,310 --> 00:30:19,860
Can I figure out
what the null space
481
00:30:19,860 --> 00:30:28,470
is for this matrix, A transpose,
by looking at the graph?
482
00:30:28,470 --> 00:30:33,070
I'm happy if I don't
have to do elimination.
483
00:30:33,070 --> 00:30:35,780
I can do elimination,
we know how to do,
484
00:30:35,780 --> 00:30:39,110
we know how to find
the null space basis.
485
00:30:39,110 --> 00:30:42,580
We can do elimination
on this matrix,
486
00:30:42,580 --> 00:30:48,150
and we'll get it into a good
reduced row echelon form,
487
00:30:48,150 --> 00:30:51,330
and the special solutions
will pop right out.
488
00:30:51,330 --> 00:30:56,040
But I would like to --
even to do it without that.
489
00:30:56,040 --> 00:30:59,820
Let me just ask you first,
if I did elimination
490
00:30:59,820 --> 00:31:06,780
on that, on that, matrix, what
would the last row become?
491
00:31:06,780 --> 00:31:10,220
What would the last row -- if I
do elimination on that matrix,
492
00:31:10,220 --> 00:31:16,380
the last row of R will
be all zeros, right?
493
00:31:16,380 --> 00:31:17,190
Why?
494
00:31:17,190 --> 00:31:19,770
Because the rank is three.
495
00:31:19,770 --> 00:31:22,720
We only going to
have three pivots.
496
00:31:22,720 --> 00:31:26,540
And the fourth row will be
all zeros when we eliminate.
497
00:31:26,540 --> 00:31:32,020
So elimination will tell us
what, what we spotted earlier,
498
00:31:32,020 --> 00:31:36,160
what's the null space -- all
the, all the information,
499
00:31:36,160 --> 00:31:38,450
what are the dependencies.
500
00:31:38,450 --> 00:31:42,790
We'll find those by elimination,
but here in a real example,
501
00:31:42,790 --> 00:31:44,830
we can find them by thinking.
502
00:31:44,830 --> 00:31:46,030
OK.
503
00:31:46,030 --> 00:31:52,320
Again, my question is,
what is a solution y?
504
00:31:52,320 --> 00:31:55,940
How could current travel
around this network
505
00:31:55,940 --> 00:32:02,440
without collecting any
charge at the nodes?
506
00:32:02,440 --> 00:32:03,650
Tell me a y.
507
00:32:03,650 --> 00:32:04,540
OK.
508
00:32:04,540 --> 00:32:12,650
So a basis for the null
space of A transpose.
509
00:32:12,650 --> 00:32:15,980
How many vectors I looking for?
510
00:32:15,980 --> 00:32:17,220
Two.
511
00:32:17,220 --> 00:32:18,750
It's a two dimensional space.
512
00:32:18,750 --> 00:32:21,350
My basis should have
two vectors in it.
513
00:32:21,350 --> 00:32:23,570
Give me one.
514
00:32:23,570 --> 00:32:24,750
One set of currents.
515
00:32:24,750 --> 00:32:28,470
Suppose, let me start it.
516
00:32:28,470 --> 00:32:31,801
Let me start with y1 as one.
517
00:32:31,801 --> 00:32:32,300
OK.
518
00:32:32,300 --> 00:32:38,860
So one unit of -- one amp
travels on edge one with
519
00:32:38,860 --> 00:32:40,120
the arrow.
520
00:32:40,120 --> 00:32:41,480
OK, then what?
521
00:32:41,480 --> 00:32:42,275
What is y2?
522
00:32:44,790 --> 00:32:47,000
It's one also, right?
523
00:32:47,000 --> 00:32:50,230
And of course what you did was
solve Kirchoff's Current Law
524
00:32:50,230 --> 00:32:52,770
quickly in the second equation.
525
00:32:52,770 --> 00:32:53,600
OK.
526
00:32:53,600 --> 00:32:57,400
Now we've got one amp leaving
node one, coming around to node
527
00:32:57,400 --> 00:32:57,900
three.
528
00:32:57,900 --> 00:32:58,990
What shall we do now?
529
00:33:01,920 --> 00:33:05,350
Well, what shall I take
for y3 in other words?
530
00:33:05,350 --> 00:33:08,870
Oh, I've got a choice, but
why not make it what you said,
531
00:33:08,870 --> 00:33:11,530
negative one.
532
00:33:11,530 --> 00:33:16,150
So I have just sent current,
one amp, around that loop.
533
00:33:19,130 --> 00:33:23,000
What shall y4 and
y5 be in this case?
534
00:33:23,000 --> 00:33:25,050
We could take them to be zero.
535
00:33:25,050 --> 00:33:31,410
This satisfies
Kirchoff's Current Law.
536
00:33:31,410 --> 00:33:36,190
We could check it patiently,
that minus y1 minus y3
537
00:33:36,190 --> 00:33:37,050
gives zero.
538
00:33:37,050 --> 00:33:38,920
We know y1 is y2.
539
00:33:38,920 --> 00:33:42,710
The others, y4 plus
y5 is certainly zero.
540
00:33:42,710 --> 00:33:46,880
Any current around
a loop satisfies --
541
00:33:46,880 --> 00:33:49,090
satisfies the Current Law.
542
00:33:49,090 --> 00:33:52,140
Now you know how
to get another one.
543
00:33:52,140 --> 00:33:53,150
OK.
544
00:33:53,150 --> 00:33:55,890
Take current around this loop.
545
00:33:55,890 --> 00:34:04,650
So now let y3 be one, y5 be
one, and y4 be minus one.
546
00:34:04,650 --> 00:34:10,400
And so, so we have the first
basis vector sent current
547
00:34:10,400 --> 00:34:13,260
around that loop, the
second basis vector
548
00:34:13,260 --> 00:34:14,500
sends current around that
549
00:34:14,500 --> 00:34:15,590
loop.
550
00:34:15,590 --> 00:34:18,219
And I've -- and those
are independent,
551
00:34:18,219 --> 00:34:23,980
and I've got two solutions --
two vectors in the null space
552
00:34:23,980 --> 00:34:28,650
of A transpose, two solutions
to Kirchoff's Current Law.
553
00:34:28,650 --> 00:34:31,590
Of course you would
say what about sending
554
00:34:31,590 --> 00:34:34,830
current around the big loop.
555
00:34:34,830 --> 00:34:36,889
What about that vector?
556
00:34:36,889 --> 00:34:44,560
One for y1, one for y2,
nothing f- on y3, one for y5,
557
00:34:44,560 --> 00:34:46,889
and minus one for y4.
558
00:34:46,889 --> 00:34:48,690
What about that?
559
00:34:48,690 --> 00:34:52,630
Is that, is that in the
null space of A transpose?
560
00:34:52,630 --> 00:34:53,860
Sure.
561
00:34:53,860 --> 00:35:01,790
So why don't we now have a
third vector in the basis?
562
00:35:01,790 --> 00:35:05,890
Because it's not
independent, right?
563
00:35:05,890 --> 00:35:07,430
It's not independent.
564
00:35:07,430 --> 00:35:10,910
This vector is the
sum of those two.
565
00:35:10,910 --> 00:35:14,200
If I send current around
that and around that --
566
00:35:14,200 --> 00:35:18,940
then on this edge y3 it's going
to cancel out and I'll have
567
00:35:18,940 --> 00:35:23,080
altogether current around
the whole, the outside loop.
568
00:35:23,080 --> 00:35:24,700
That's what this
one is, but it's
569
00:35:24,700 --> 00:35:28,240
a combination of those two.
570
00:35:28,240 --> 00:35:33,300
Do you see that I've now, I've
identified the null space of A
571
00:35:33,300 --> 00:35:36,260
transpose --
572
00:35:36,260 --> 00:35:40,640
but more than that, we've
solved Kirchoff's Current Law.
573
00:35:43,160 --> 00:35:48,590
And understood it in
terms of the network.
574
00:35:48,590 --> 00:35:49,180
OK.
575
00:35:49,180 --> 00:35:53,010
So that's the null
space of A transpose.
576
00:35:53,010 --> 00:35:58,030
I guess I -- there's always one
more space to ask you about.
577
00:35:58,030 --> 00:36:04,370
Let's see, I guess I need the
row space of A, the column
578
00:36:04,370 --> 00:36:05,440
space of A transpose.
579
00:36:10,550 --> 00:36:14,240
So what's N, what's
its dimension?
580
00:36:14,240 --> 00:36:15,800
Yup?
581
00:36:15,800 --> 00:36:18,450
What's the dimension
of the row space of A?
582
00:36:18,450 --> 00:36:21,672
If I look at the original
A, it had five rows.
583
00:36:21,672 --> 00:36:22,755
How many were independent?
584
00:36:27,220 --> 00:36:30,620
Oh, I guess I'm asking
you the rank again, right?
585
00:36:30,620 --> 00:36:33,490
And the answer is three, right?
586
00:36:33,490 --> 00:36:35,610
Three independent rows.
587
00:36:35,610 --> 00:36:38,760
When I transpose it, there's
three independent columns.
588
00:36:38,760 --> 00:36:42,650
Are those columns
independent, those three?
589
00:36:42,650 --> 00:36:45,620
The first three columns,
are they the pivot columns
590
00:36:45,620 --> 00:36:46,720
of the matrix?
591
00:36:46,720 --> 00:36:48,180
No.
592
00:36:48,180 --> 00:36:51,450
Those three columns
are not independent.
593
00:36:51,450 --> 00:36:56,000
There's a in fact, this tells
me a relation between them.
594
00:36:56,000 --> 00:36:57,940
There's a vector in
the null space that
595
00:36:57,940 --> 00:37:01,530
says the first column
plus the second column
596
00:37:01,530 --> 00:37:03,400
equals the third column.
597
00:37:03,400 --> 00:37:07,490
They're not independent
because they come from a loop.
598
00:37:07,490 --> 00:37:11,570
So the pivot columns, the
pivot columns of this matrix
599
00:37:11,570 --> 00:37:18,630
will be the first, the second,
not the third, but the fourth.
600
00:37:18,630 --> 00:37:24,360
One, columns one,
two, and four are OK.
601
00:37:24,360 --> 00:37:28,170
Where are they -- those are
the columns of A transpose,
602
00:37:28,170 --> 00:37:30,350
those correspond to edges.
603
00:37:30,350 --> 00:37:35,340
So there's edge one,
there's edge two,
604
00:37:35,340 --> 00:37:37,600
and there's edge four.
605
00:37:42,370 --> 00:37:46,930
So there's a -- that's like --
606
00:37:46,930 --> 00:37:49,040
is a, smaller graph.
607
00:37:49,040 --> 00:37:52,860
If I just look at the part
of the graph that I've, that
608
00:37:52,860 --> 00:37:56,610
I've, thick -- used
with thick edges,
609
00:37:56,610 --> 00:38:00,240
it has the same four nodes.
610
00:38:00,240 --> 00:38:03,400
It only has three edges.
611
00:38:03,400 --> 00:38:08,520
And the, those edges correspond
to the independent guys.
612
00:38:08,520 --> 00:38:14,970
And in the graph there --
those three edges have no loop,
613
00:38:14,970 --> 00:38:15,900
right?
614
00:38:15,900 --> 00:38:19,550
The independent ones are the
ones that don't have a loop.
615
00:38:19,550 --> 00:38:22,820
All the -- dependencies
came from loops.
616
00:38:22,820 --> 00:38:25,800
They were the things in the
null space of A transpose.
617
00:38:25,800 --> 00:38:28,140
If I take three
pivot columns, there
618
00:38:28,140 --> 00:38:31,620
are no dependencies among
them, and they form a graph
619
00:38:31,620 --> 00:38:34,630
without a loop, and I
just want to ask you
620
00:38:34,630 --> 00:38:37,640
what's the name for a
graph without a loop?
621
00:38:37,640 --> 00:38:43,590
So a graph without a loop is
-- has got not very many edges,
622
00:38:43,590 --> 00:38:44,520
right?
623
00:38:44,520 --> 00:38:48,180
I've got four nodes and
it only has three edges,
624
00:38:48,180 --> 00:38:53,360
and if I put another edge
in, I would have a loop.
625
00:38:53,360 --> 00:38:56,390
So it's this graph
with no loops,
626
00:38:56,390 --> 00:39:01,670
and it's the one where the
rows of A are independent.
627
00:39:01,670 --> 00:39:04,400
And what's a graph
called that has no loops?
628
00:39:04,400 --> 00:39:06,810
It's called a tree.
629
00:39:06,810 --> 00:39:11,980
So a tree is the name for
a graph with no loops.
630
00:39:17,290 --> 00:39:24,400
And just to take
one last step here.
631
00:39:24,400 --> 00:39:28,170
Using our formula for dimension.
632
00:39:28,170 --> 00:39:33,710
Using our formula for
dimension, let's look --
633
00:39:33,710 --> 00:39:41,370
once at this formula.
634
00:39:41,370 --> 00:39:49,560
The dimension of the null
space of A transpose is m-r.
635
00:39:49,560 --> 00:39:51,230
OK.
636
00:39:51,230 --> 00:39:58,720
This is the number of loops,
number of independent loops.
637
00:39:58,720 --> 00:40:00,300
m is the number of edges.
638
00:40:04,250 --> 00:40:05,210
And what is r?
639
00:40:08,780 --> 00:40:12,460
What is r for our -- we'll
have to remember way back.
640
00:40:12,460 --> 00:40:17,970
The rank came -- from looking
at the columns of our matrix.
641
00:40:17,970 --> 00:40:19,950
So what's the rank?
642
00:40:19,950 --> 00:40:21,130
Let's just remember.
643
00:40:21,130 --> 00:40:26,710
Rank was -- you remember
there was one --
644
00:40:26,710 --> 00:40:29,750
we had a one dimensional
-- rank was n minus one,
645
00:40:29,750 --> 00:40:33,690
that's what I'm
struggling to say.
646
00:40:33,690 --> 00:40:37,810
Because there were n columns
coming from the n nodes,
647
00:40:37,810 --> 00:40:45,030
so it's minus, the number
of nodes minus one,
648
00:40:45,030 --> 00:40:50,070
because of that C, that
one one one one vector
649
00:40:50,070 --> 00:40:51,920
in the null space.
650
00:40:51,920 --> 00:40:54,210
The columns were
not independent.
651
00:40:54,210 --> 00:40:58,190
There was one dependency,
so we needed n minus one.
652
00:40:58,190 --> 00:41:01,610
This is a great formula.
653
00:41:01,610 --> 00:41:06,230
This is like the
first shall I, --
654
00:41:06,230 --> 00:41:09,410
write it slightly differently?
655
00:41:09,410 --> 00:41:14,890
The number of edges --
let me put things --
656
00:41:14,890 --> 00:41:17,940
have I got it right?
657
00:41:17,940 --> 00:41:22,630
Number of edges is m, the
number -- r- is m-r, OK.
658
00:41:22,630 --> 00:41:24,500
So, so I'm getting --
659
00:41:24,500 --> 00:41:27,070
let me put the number of
nodes on the other side.
660
00:41:27,070 --> 00:41:31,130
So I -- the number of nodes --
661
00:41:31,130 --> 00:41:34,890
I'll move that to
the other side --
662
00:41:34,890 --> 00:41:46,320
minus the number of edges
plus the number of loops is --
663
00:41:46,320 --> 00:41:50,060
I have minus, minus one is one.
664
00:41:50,060 --> 00:41:52,190
The number of nodes
minus the number
665
00:41:52,190 --> 00:41:56,030
of edges plus the
number of loops is one.
666
00:41:56,030 --> 00:41:58,460
These are like zero
dimensional guys.
667
00:41:58,460 --> 00:42:01,230
They're the points on the graph.
668
00:42:01,230 --> 00:42:03,570
The edges are like one
dimensional things,
669
00:42:03,570 --> 00:42:06,480
they're, they connect nodes.
670
00:42:06,480 --> 00:42:09,420
The loops are like two
dimensional things.
671
00:42:09,420 --> 00:42:12,240
They have, like, an area.
672
00:42:12,240 --> 00:42:16,030
And this count works
for every graph.
673
00:42:16,030 --> 00:42:22,950
And it's known as
Euler's Formula.
674
00:42:22,950 --> 00:42:26,770
We see Euler again,
that guy never stopped.
675
00:42:26,770 --> 00:42:28,940
OK.
676
00:42:28,940 --> 00:42:33,400
And can we just check
-- so what I saying?
677
00:42:33,400 --> 00:42:37,340
I'm saying that linear algebra
proves Euler's Formula.
678
00:42:37,340 --> 00:42:43,650
Euler's Formula is this great
topology fact about any graph.
679
00:42:43,650 --> 00:42:45,660
I'll draw, let me
draw another graph,
680
00:42:45,660 --> 00:42:51,980
let me draw a graph with
more edges and loops.
681
00:42:51,980 --> 00:42:53,610
Let me put in lots of --
682
00:42:53,610 --> 00:42:54,530
OK.
683
00:42:54,530 --> 00:42:56,500
I just drew a graph there.
684
00:42:56,500 --> 00:42:58,990
So what are the, what are the
quantities in that formula?
685
00:42:58,990 --> 00:43:00,870
How many nodes have I got?
686
00:43:00,870 --> 00:43:01,630
Looks like five.
687
00:43:04,180 --> 00:43:05,730
How many edges have I got?
688
00:43:05,730 --> 00:43:10,700
One two three four
five six seven.
689
00:43:10,700 --> 00:43:12,150
How many loops have I got?
690
00:43:12,150 --> 00:43:15,010
One two three.
691
00:43:15,010 --> 00:43:19,810
And Euler's right,
I always get one.
692
00:43:19,810 --> 00:43:27,700
That, this formula, is extremely
useful in understanding
693
00:43:27,700 --> 00:43:31,560
the relation of these quantities
-- the number of nodes,
694
00:43:31,560 --> 00:43:34,780
the number of edges,
and the number of loops.
695
00:43:34,780 --> 00:43:35,970
OK.
696
00:43:35,970 --> 00:43:39,890
Just complete this
lecture by completing
697
00:43:39,890 --> 00:43:42,310
this picture, this cycle.
698
00:43:42,310 --> 00:43:45,320
So let me come to the --
699
00:43:50,610 --> 00:43:57,200
so this expresses the
equations of applied math.
700
00:43:57,200 --> 00:43:59,800
This, let me call these
potential differences,
701
00:43:59,800 --> 00:44:04,100
say, E. So E is A x.
702
00:44:04,100 --> 00:44:08,010
That's the equation
for this step.
703
00:44:08,010 --> 00:44:11,890
The currents come from
the potential differences.
704
00:44:11,890 --> 00:44:15,020
y is C E.
705
00:44:15,020 --> 00:44:20,230
The potential -- the currents
satisfy Kirchoff's Current Law.
706
00:44:20,230 --> 00:44:23,230
Those are the equations of --
707
00:44:23,230 --> 00:44:25,290
with no source terms.
708
00:44:25,290 --> 00:44:32,590
Those are the equations of
electrical circuits of many --
709
00:44:32,590 --> 00:44:37,940
those are like the, the
most basic three equations.
710
00:44:37,940 --> 00:44:41,110
Applied math comes
in this structure.
711
00:44:41,110 --> 00:44:43,690
The only thing I haven't
got yet in the picture
712
00:44:43,690 --> 00:44:49,260
is an outside source to
make something happen.
713
00:44:49,260 --> 00:44:52,650
I could add a
current source here,
714
00:44:52,650 --> 00:44:55,610
I could, I could add
external currents
715
00:44:55,610 --> 00:44:57,370
going in and out of nodes.
716
00:44:57,370 --> 00:44:59,790
I could add batteries
in the edges.
717
00:44:59,790 --> 00:45:01,530
Those are two ways.
718
00:45:01,530 --> 00:45:05,860
If I add batteries in the edges,
they, they come into here.
719
00:45:05,860 --> 00:45:07,650
Let me add current sources.
720
00:45:07,650 --> 00:45:13,160
If I add current sources,
those come in here.
721
00:45:13,160 --> 00:45:16,030
So there's a, there's
where current sources go,
722
00:45:16,030 --> 00:45:22,160
because the F is a like a
current coming from outside.
723
00:45:22,160 --> 00:45:25,080
So we have our edges,
we have our graph,
724
00:45:25,080 --> 00:45:33,370
and then I send one amp into
this node and out of this node
725
00:45:33,370 --> 00:45:36,970
-- and that gives
me, a right-hand side
726
00:45:36,970 --> 00:45:38,840
in Kirchoff's Current Law.
727
00:45:38,840 --> 00:45:41,290
And can I -- to
complete the lecture,
728
00:45:41,290 --> 00:45:45,420
I'm just going to put these
three equations together.
729
00:45:45,420 --> 00:45:49,240
So I start with x, my unknown.
730
00:45:49,240 --> 00:45:51,350
I multiply by A.
731
00:45:51,350 --> 00:45:53,520
That gives me the
potential differences.
732
00:45:53,520 --> 00:45:57,100
That was our matrix A that
the whole thing started with.
733
00:45:57,100 --> 00:45:59,720
I multiply by C.
734
00:45:59,720 --> 00:46:03,280
Those are the physical
constants in Ohm's Law.
735
00:46:03,280 --> 00:46:05,500
Now I have y.
736
00:46:05,500 --> 00:46:12,270
I multiply y by A
transpose, and now I have F.
737
00:46:12,270 --> 00:46:16,660
So there is the whole thing.
738
00:46:16,660 --> 00:46:22,870
There's the basic
equation of applied math.
739
00:46:22,870 --> 00:46:28,700
Coming from these three steps,
in which the last step is
740
00:46:28,700 --> 00:46:30,140
this balance equation.
741
00:46:30,140 --> 00:46:33,560
There's always a balance
equation to look for.
742
00:46:33,560 --> 00:46:34,880
These are the --
743
00:46:34,880 --> 00:46:37,350
when I say the most basic
equations of applied
744
00:46:37,350 --> 00:46:37,980
mathematics --
745
00:46:37,980 --> 00:46:41,500
I should say, in equilibrium.
746
00:46:41,500 --> 00:46:43,820
Time isn't in this problem.
747
00:46:43,820 --> 00:46:47,800
I'm not -- and Newton's
Law isn't acting here.
748
00:46:47,800 --> 00:46:51,230
I'm, I'm looking at the --
equations when everything has
749
00:46:51,230 --> 00:46:54,110
settled down, how do
the currents distribute
750
00:46:54,110 --> 00:46:55,780
in the network.
751
00:46:55,780 --> 00:47:00,480
And of course there are
big codes to solve the --
752
00:47:00,480 --> 00:47:04,860
this is the basic problem
of numerical linear algebra
753
00:47:04,860 --> 00:47:09,190
for systems of equations,
because that's how they come.
754
00:47:09,190 --> 00:47:13,320
And my final question.
755
00:47:13,320 --> 00:47:18,940
What can you tell me about
this matrix A transpose C A?
756
00:47:18,940 --> 00:47:21,400
Or even A transpose A?
757
00:47:21,400 --> 00:47:24,310
I'll just close
with that question.
758
00:47:24,310 --> 00:47:28,280
What do you know about
the matrix A transpose A?
759
00:47:28,280 --> 00:47:32,400
It is always symmetric, right.
760
00:47:32,400 --> 00:47:33,350
OK, thank.
761
00:47:33,350 --> 00:47:38,590
So I'll see you Wednesday for a
full review of these chapters,
762
00:47:38,590 --> 00:47:41,230
and Friday you get to tell me.
763
00:47:41,230 --> 00:47:42,780
Thanks.